nLab universal central extension

Definition

For any algebraic object, such as an associative algebra, group, Lie algebra, etc. such that there is a notion of central extension, the universal central extension is, if it exists, the initial object in the category of central extensions of that object.

Examples

For $G$ a discrete group which is perfect, the universal central extension of $G$ is the essentially unique group $K$ that is a Schur covering group? of $G$.

Properties

• The universal central extension of a perfect group is also perfect.

• The universal central extension of a perfect group is a Schur-trivial group, and hence a superperfect group (superperfect means that it’s perfect and Schur-trivial).

• The universal central extension operator is idempotent, i.e., the universal central extension of the universal central extension is the universal central extension. This follows directly from the universal central extension being a Schur-trivial group.

Related concepts

• maximal central extension

• universal higher central extension

References

General

Discussion in semi-abelian categories:

• Jose Casas, Tim Van der Linden, A relative theory of universal central extensions (arXiv:0908.3762)

For discrete groups

• Groupprops, Universal central extension

For Lie groups

• Karl-Hermann Neeb, Universal Central Extensions of Lie Groups, Acta Applicandae Mathematicae (2002) 73: 175 (doi:10.1023/A:1019743224737)

For super Lie algebras

For super Lie algebras

• Erhard Neher, An introduction to universal central extensions of Lie superalgebras (pdf)